Effects of the spherical terrain on gravity and the geoid

Abstract

The determination of the gravimetric geoid is based on the magnitude of gravity observed at the topographical surface and applied in two boundary value problems of potential theory: the Dirichlet problem (for downward continuation of gravity anomalies from the topography to the geoid) and the Stokes problem (for transformation of gravity anomalies into the disturbing gravity potential at the geoid). Since both problems require involved functions to be harmonic everywhere outside the geoid, proper reduction of gravity must be applied. This contribution deals with far-zone effects of the global terrain on gravity and the geoid in the Stokes–Helmert scheme. A spherical harmonic model of the global topography and a Molodenskij-type spectral approach are used for a derivation of suitable computational formulae. Numerical results for a part of the Canadian Rocky Mountains are presented to illustrate the significance of these effects in precise (i.e. centimetre) geoid computations. Their omission can be responsible for a long-frequency bias in the geoid, especially over mountainous areas. Due to the rough topography of the testing area, these numerical values can be used as maximum global estimates of the effects (maybe with the exception of the Himalayas). This study is a continuation of efforts to model adequately the topographical effects on gravity and the geoid, especially of a comparing the effects of the planar topographical plate and the spherical topographical shell on gravity and the geoid [Vanicek, Novak, Martinec (2001) J Geod 75: 210–215].